{"id":34248,"date":"2019-11-27T12:41:39","date_gmt":"2019-11-27T15:41:39","guid":{"rendered":"https:\/\/blog.estrategiavestibulares.com.br\/?p=34248"},"modified":"2022-07-08T14:57:58","modified_gmt":"2022-07-08T17:57:58","slug":"prova-matematica-fuvest-2020","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/prova-matematica-fuvest-2020\/","title":{"rendered":"Prova de matem\u00e1tica Fuvest 2020 com Resolu\u00e7\u00e3o Comentada"},"content":{"rendered":"<p>Ol&aacute;, pessoal&hellip; Tudo bem? Sou o prof. Mar&ccedil;al, do Estrat&eacute;gia Vestibulares, e escrevo este artigo para resolver as quest&otilde;es da prova da FUVEST 2020, da disciplina de Matem&aacute;tica. Nesta p&aacute;gina, voc&ecirc; vai conferir a resolu&ccedil;&atilde;o completa e ainda vai poder baixar gratuitamente os coment&aacute;rios em PDF. Vamos nessa??<div class=\"wp-block-file\">\"&gt;Prova FUVEST 2020 &ndash; Matem&aacute;tica &ndash; Resolu&ccedil;&atilde;o\" class=\"wp-block-file__button\" download&gt;Baixar<\/div><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/prova-matematica-fuvest-2020\/#Prova-FUVEST-2020\" >Prova FUVEST 2020<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/prova-matematica-fuvest-2020\/#Veja-tambem\" >Veja tamb&eacute;m:<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-prova-fuvest-2020\"><span class=\"ez-toc-section\" id=\"Prova-FUVEST-2020\"><\/span>Prova FUVEST 2020<span class=\"ez-toc-section-end\"><\/span><\/h2><h3 class=\"wp-block-heading\" id=\"h-questao-13\"><strong>Quest&atilde;o 13<\/strong><\/h3><p>Carros que saem da cidade A rumo a alguma das cidades tur&iacute;sticas E, F e G fazem caminhos diversos, passando por pelo menos uma das cidades B, C e D, apenas no sentido indicado pelas setas, como mostra a figura. Os n&uacute;meros indicados nas setas s&atilde;o as probabilidades, dentre esses carros, de se ir de uma cidade a outra.<\/p><div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/11\/image215.png\" alt=\"\" class=\"wp-image-34251\"><\/figure><\/div><p>Nesse cen&aacute;rio, a probabilidade de um carro ir de A a\nF &eacute;<\/p><p>(A) 0,120.<\/p><p>(B) 0,216.<\/p><p>(C) 0,264.<\/p><p>(D) 0,336.<\/p><p>(E) 0,384.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>Para ir de <em>A<\/em> a <em>F<\/em>, h&aacute; tr&ecirc;s caminhos poss&iacute;veis: <em>ACF<\/em>, <em>ABCF<\/em> e <em>ABDF<\/em>. <br>Em cada caminho, as probabilidades s&atilde;o dadas pelos produtos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;ACF%5Crightarrow&amp;space;0,2%5Ccdot&amp;space;0,6=0,12\" alt=\"\\large ACF\\rightarrow 0,2\\cdot 0,6=0,12\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;ABCF%5Crightarrow&amp;space;0,8%5Ccdot&amp;space;0,1%5Ccdot&amp;space;0,6=0,048\" alt=\"\\large ABCF\\rightarrow 0,8\\cdot 0,1\\cdot 0,6=0,048\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;ABDF%5Crightarrow&amp;space;0,8%5Ccdot&amp;space;0,9%5Ccdot&amp;space;0,3=&amp;space;0,216\" alt=\"\\large ABDF\\rightarrow 0,8\\cdot 0,9\\cdot 0,3= 0,216\" align=\"absmiddle\"><\/p><p>Como os caminhos s&atilde;o todos alternativos, a probabilidade total de o carro ir de <em>A<\/em> a <em>F<\/em> &eacute; dada pela soma<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;ACF+ABCF&amp;space;+ABDF\" alt=\"\\large ACF+ABCF +ABDF\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;0,12+0,048+0,216\" alt=\"\\large 0,12+0,048+0,216\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;0,384\" alt=\"\\large 0,384\" align=\"absmiddle\"><\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: E<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-14\"><strong>Quest&atilde;o 14<\/strong><\/h3><p>Se, em 15 anos, o sal&aacute;rio m&iacute;nimo teve um aumento\nnominal de 300% e a infla&ccedil;&atilde;o foi de 100%, &eacute; correto afirmar que o aumento real\ndo sal&aacute;rio m&iacute;nimo, nesse per&iacute;odo, foi de<\/p><p>(A) 50%.<\/p><p>(B) 100%.<\/p><p>(C) 150%.<\/p><p>(D) 200%.<\/p><p>(E) 250%.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-1\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>Considerando o sal&aacute;rio x com um crescimento de 300%, ou seja, de 3x, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sal%5C'ario=x%5Crightarrow&amp;space;x+3x=4x\" alt=\"\\large sal\\'ario=x\\rightarrow x+3x=4x\" align=\"absmiddle\"><\/p><p>J&aacute; seu poder de compra, sofrendo uma alta de 100%, ou seja, de 1x, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;compra=x%5Crightarrow&amp;space;x+1x=2x\" alt=\"\\large compra=x\\rightarrow x+1x=2x\" align=\"absmiddle\"><\/p><p>Desse modo, podemos perceber que o sal&aacute;rio que tinha uma rela&ccedil;&atilde;o de 1 para 1 com a compra, agora &eacute; o dobro da compra. Se o poder de compra &eacute; o dobro, sofreu aumento de 100%, indicando o gabarito b).<\/p><p>Podemos, alternativamente, utilizar a f&oacute;rmula para o ganho real:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1+A=%5Cleft(1+i%5Cright)%5Ccdot%5Cleft(1+R%5Cright)\" alt=\"\\large 1+A=\\left(1+i\\right)\\cdot\\left(1+R\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1+3=%5Cleft(1+1%5Cright)%5Ccdot%5Cleft(1+R%5Cright)\" alt=\"\\large 1+3=\\left(1+1\\right)\\cdot\\left(1+R\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;4=2%5Ccdot%5Cleft(1+R%5Cright)\" alt=\"\\large 4=2\\cdot\\left(1+R\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2=1+R\" alt=\"\\large 2=1+R\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;1=R\" alt=\"\\large 1=R\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;100%5C%=R\" alt=\"\\large 100\\%=R\" align=\"absmiddle\"><\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: B<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-15\"><strong>Quest&atilde;o 15<\/strong><\/h3><p>O cilindro de papel&atilde;o central de uma fita crepe tem\nraio externo de 3 cm. A fita tem espessura de 0,01 cm e d&aacute; 100 voltas\ncompletas. Considerando que, a cada volta, o raio externo do rolo &eacute; aumentado\nno valor da espessura da fita, o comprimento total da fita &eacute; de,\naproximadamente,<\/p><div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/11\/image216.png\" alt=\"\" class=\"wp-image-34257\"><\/figure><\/div><p>(A) 9,4 m.<\/p><p>(B) 11,0 m.<\/p><p>(C) 18,8 m.<\/p><p>(D) 22,0 m.<\/p><p>(E) 25,1 m.<\/p><p><strong>Note e Adote:<\/strong><br>&pi; = 3,14<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-2\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>Sabendo que o comprimento de uma circunfer&ecirc;ncia &eacute; dado por <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C=2%5Ccdot%5Cpi%5Ccdot&amp;space;R\" alt=\"\\large C=2\\cdot\\pi\\cdot R\" align=\"absmiddle\"> , temos a seguinte sequ&ecirc;ncia de comprimentos ao enrolar a fita no cilintro:<\/p><p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_1=2%5Ccdot%5Cpi%5Ccdot&amp;space;R_1\" alt=\"\\large C_1=2\\cdot\\pi\\cdot R_1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_2=2%5Ccdot%5Cpi%5Ccdot&amp;space;R_2\" alt=\"\\large C_2=2\\cdot\\pi\\cdot R_2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_3=2%5Ccdot%5Cpi%5Ccdot&amp;space;R_3%5C%5C\" alt=\"\\large C_3=2\\cdot\\pi\\cdot R_3\\\\\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cvdots\" alt=\"\\large \\vdots\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_%7B100%7D=2%5Ccdot%5Cpi%5Ccdot&amp;space;R_%7B100%7D\" alt=\"\\large C_{100}=2\\cdot\\pi\\cdot R_{100}\" align=\"absmiddle\"><\/p><p>Dessa forma, o comprimento total da fita &eacute; dado por:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_T=C_1+C_2+C_3+%5Cldots+C_%7B100%7D\" alt=\"\\large C_T=C_1+C_2+C_3+\\ldots+C_{100}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_T=2%5Ccdot%5Cpi%5Ccdot&amp;space;R_1+2%5Ccdot%5Cpi%5Ccdot&amp;space;R_2+2%5Ccdot%5Cpi%5Ccdot&amp;space;R_3+%5Cldots+2%5Ccdot%5Cpi%5Ccdot&amp;space;R_%7B100%7D\" alt=\"\\large C_T=2\\cdot\\pi\\cdot R_1+2\\cdot\\pi\\cdot R_2+2\\cdot\\pi\\cdot R_3+\\ldots+2\\cdot\\pi\\cdot R_{100}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_T=2%5Ccdot%5Cpi%5Ccdot%5Cleft(R_1+R_2+R_3+%5Cldots+R_%7B100%7D%5Cright)\" alt=\"\\large C_T=2\\cdot\\pi\\cdot\\left(R_1+R_2+R_3+\\ldots+R_{100}\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_T=2%5Ccdot3,14%5Ccdot%5Cleft(3,01+3,02+3,03+%5Cldots+4%5Cright)\" alt=\"\\large C_T=2\\cdot3,14\\cdot\\left(3,01+3,02+3,03+\\ldots+4\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_T=2%5Ccdot3,14%5Ccdot%5Cfrac%7B%5Cleft(3,01+4%5Cright)%5Ccdot100%7D%7B2%7D\" alt=\"\\large C_T=2\\cdot3,14\\cdot\\frac{\\left(3,01+4\\right)\\cdot100}{2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_T=3,14%5Ccdot7,01%5Ccdot100\" alt=\"\\large C_T=3,14\\cdot7,01\\cdot100\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_T=22,0114%5Ccdot100%5C&amp;space;cm\" alt=\"\\large C_T=22,0114\\cdot100\\ cm\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C_T=22,0114%5C&amp;space;%5C&amp;space;m\" alt=\"\\large C_T=22,0114\\ \\ m\" align=\"absmiddle\"><\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: D<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-16\"><strong>Quest&atilde;o 16<\/strong><\/h3><p>Um objeto &eacute; formado por 4 hastes r&iacute;gidas conectadas em seus extremos por articula&ccedil;&otilde;es, cujos centros s&atilde;o os v&eacute;rtices de um paralelogramo. As hastes movimentam-se de tal forma que o paralelogramo permanece sempre no mesmo plano. A cada configura&ccedil;&atilde;o desse objeto, associa-se d, a medida do menor &acirc;ngulo interno do paralelogramo. A &aacute;rea da regi&atilde;o delimitada pelo paralelogramo quando d = 90&deg; &eacute; A.<\/p><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/11\/image217.png\" alt=\"\" class=\"wp-image-34259\"><\/figure><p>Para que a &aacute;rea da regi&atilde;o delimitada pelo paralelogramo seja <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7BA%7D%7B2%7D\" alt=\"\\large \\frac{A}{2}\" align=\"absmiddle\"> o valor de <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Ctheta\" alt=\"\\large \\theta\" align=\"absmiddle\"> &eacute; necessariamente, igual a:<\/p><p>(A) 15&deg;<\/p><p>(B) 22,5&deg;<\/p><p>(C) 30&deg;<\/p><p>(D) 45&deg;<\/p><p>(E) 60&deg;<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-3\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>Chamando os lados do ret&acirc;ngulo de x e y, temos a seguinte rela&ccedil;&atilde;o entre suas &aacute;reas:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5Ccdot&amp;space;y%5Ccdot&amp;space;sen%5Cleft(%5Ctheta%5Cright)=%5Cfrac%7B1%7D%7B2%7D%5Ccdot&amp;space;x%5Ccdot&amp;space;y\" alt=\"\\large x\\cdot y\\cdot sen\\left(\\theta\\right)=\\frac{1}{2}\\cdot x\\cdot y\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;sen%5Cleft(%5Ctheta%5Cright)=%5Cfrac%7B1%7D%7B2%7D\" alt=\"\\large sen\\left(\\theta\\right)=\\frac{1}{2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Ctheta=30%5Cdegree\" alt=\"\\large \\theta=30\\degree\" align=\"absmiddle\"><\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: C<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-17\"><strong>Quest&atilde;o 17<\/strong><\/h3><p>A menor esfera na qual um paralelep&iacute;pedo\nreto-ret&acirc;ngulo de medidas 7 cm x 4 cm x 4 cm est&aacute; inscrito tem di&acirc;metro de<\/p><p>(A) 9 cm.<\/p><p>(B) 10 cm.<\/p><p>(C) 11 cm.<\/p><p>(D) 12 cm.<\/p><p>(E) 15 cm.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-4\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>Para que a esfera tangencie todos os v&eacute;rtices do paralelep&iacute;pedo, seu di&acirc;metro ser&aacute; igual &agrave; diagonal do paralelep&iacute;pedo.<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_P%5E2=4%5E2+4%5E2+7%5E2\" alt=\"\\large d_P^2=4^2+4^2+7^2\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_P%5E2=16+16+49\" alt=\"\\large d_P^2=16+16+49\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_P%5E2=81\" alt=\"\\large d_P^2=81\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_P=%5Csqrt%7B81%7D\" alt=\"\\large d_P=\\sqrt{81}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_P=9\" alt=\"\\large d_P=9\" align=\"absmiddle\"><\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: A<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-18\"><strong>Quest&atilde;o 18<\/strong><\/h3><p>A dona de uma lanchonete observou que, vendendo um\ncombo a R$ 10,00, 200 deles s&atilde;o vendidos por dia, e que, para cada redu&ccedil;&atilde;o de\nR$ 1,00 nesse pre&ccedil;o, ela vende 100 combos a mais. Nessas condi&ccedil;&otilde;es, qual &eacute; a\nm&aacute;xima arrecada&ccedil;&atilde;o di&aacute;ria que ela espera obter com a venda desse combo?<\/p><p>(A) R$ 2.000,00<\/p><p>(B) R$ 3.200,00<\/p><p>(C) R$ 3.600,00<\/p><p>(D) R$4.000,00<\/p><p>(E) R$ 4.800,00<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-5\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>A arrecada&ccedil;&atilde;o &eacute; dada pelo produto entre o pre&ccedil;o de venda e o n&uacute;mero de vendas.<\/p><p>Arrecada&ccedil;&atilde;o = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P%5Ccdot&amp;space;V=10%5Ccdot&amp;space;200\" alt=\"\\large P\\cdot V=10\\cdot 200\" align=\"absmiddle\"><\/p><p>Como, ao retirar 1 real, ganha-se 100 clientes, ao retirar x reais do pre&ccedil;o, ganhar-se-&aacute; 100x clientes. Dessa forma, podemos reescrever nossa arrecada&ccedil;&atilde;o como:<\/p><p>Arrecada&ccedil;&atilde;o = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft(10-x%5Cright)%5Ccdot%5Cleft(200+100x%5Cright)\" alt=\"\\large \\left(10-x\\right)\\cdot\\left(200+100x\\right)\" align=\"absmiddle\"><\/p><p>Arrecada&ccedil;&atilde;o = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2000+1000x-200x-100x%C2%B2\" alt=\"\\large 2000+1000x-200x-100x&sup2;\" align=\"absmiddle\"><\/p><p>Arrecada&ccedil;&atilde;o = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;-100x%5E2+800+x+2000\" alt=\"\\large -100x^2+800+x+2000\" align=\"absmiddle\"><\/p><p>Podemos perceber que a arrecada&ccedil;&atilde;o depende da vari&aacute;vel x de forma quadr&aacute;tica, cuja par&aacute;bola apresenta concavidade negativa (a&lt;0). Assim, ao tentar maximizar a arrecada&ccedil;&atilde;o, estamos, na verdade, procurando o v&eacute;rtice da par&aacute;bola.<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5CDelta&amp;space;=8002-4%5Ccdot-100%5Ccdot2000=\" alt=\"\\large \\Delta =8002-4\\cdot-100\\cdot2000=\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;640%5C&amp;space;000+800%5C&amp;space;000=1%5C&amp;space;440%5C&amp;space;000\" alt=\"\\large 640\\ 000+800\\ 000=1\\ 440\\ 000\" align=\"absmiddle\"><\/p><p>Arrecada&ccedil;&atilde;o = <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;-%5Cfrac%7B%5CDelta&amp;space;%7D%7B4a%7D=-%5Cfrac%7B1%5C&amp;space;440%5C&amp;space;000%7D%7B4%5Ccdot(-100)%7D=%5Cfrac%7B14%5C&amp;space;400%7D%7B4%7D=&amp;space;3%5C&amp;space;600\" alt=\"\\large -\\frac{\\Delta }{4a}=-\\frac{1\\ 440\\ 000}{4\\cdot(-100)}=\\frac{14\\ 400}{4}= 3\\ 600\" align=\"absmiddle\"><\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: C<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-19\"><strong>Quest&atilde;o 19<\/strong><\/h3><p>A fun&ccedil;&atilde;o E de Euler determina, para cada n&uacute;mero\nnatural n, a quantidade de n&uacute;meros naturais menores do que n cujo m&aacute;ximo\ndivisor comum com n &eacute; igual a 1. Por exemplo, E (6) = 2 pois os n&uacute;meros menores\ndo que 6 com tal propriedade s&atilde;o 1 e 5. Qual o valor m&aacute;ximo de E (n), para n de\n20 a 25?<\/p><p>(A) 19<\/p><p>(B) 20<\/p><p>(C) 22<\/p><p>(D) 24<\/p><p>(E) 25<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-6\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>A descri&ccedil;&atilde;o da fun&ccedil;&atilde;o de Euler indica que devemos selecionar, a cada n&uacute;mero n, todos os n&uacute;meros menores que n, que sejam primos com n.<\/p><p>Para n=20, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;20%5Crightarrow1,3,7,9,11,13,17,19%5Crightarrow&amp;space;E%5Cleft(20%5Cright)=8\" alt=\"\\large 20\\rightarrow1,3,7,9,11,13,17,19\\rightarrow E\\left(20\\right)=8\" align=\"absmiddle\"><\/p><p>Para n=21, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;21%5Crightarrow1,2,4,5,8,10,11,13,16,17,19,20%5Crightarrow&amp;space;E%5Cleft(21%5Cright)=12\" alt=\"\\large 21\\rightarrow1,2,4,5,8,10,11,13,16,17,19,20\\rightarrow E\\left(21\\right)=12\" align=\"absmiddle\"><\/p><p>Perceba que, por serem n&uacute;meros compostos, acabamos por retirar todos os n&uacute;meros menores que n que tenham algum primo em comum com n.<\/p><p>Dessa forma, n&atilde;o h&aacute; necessidade de fazermos o processo para todos os n&uacute;meros do exerc&iacute;cio (entre 20 e 25), pois, nesse intervalo, s&oacute; h&aacute; um primo, o 23.<\/p><p>Assim, para o 23, n&atilde;o precisaremos retirar n&uacute;mero algum da sequ&ecirc;ncia dos n&uacute;meros menores que ele, pois &eacute; primo, resultando no maior resultado poss&iacute;vel, nesse intervalo, para E(n).<\/p><p>Portanto, n=23<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;23%5Crightarrow1,2,3,4,5,6,7,8,9,10,11,12,\" alt=\"\\large 23\\rightarrow1,2,3,4,5,6,7,8,9,10,11,12,\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;13,14,15,16,17,18,19,20,21,22%5Crightarrow22\" alt=\"\\large 13,14,15,16,17,18,19,20,21,22\\rightarrow22\" align=\"absmiddle\"><\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: C<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-20\"><strong>Quest&atilde;o 20<\/strong><\/h3><p>Se <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3x%5E2-9x+7=%5Cleft(x-a%5Cright)%5E3-%5Cleft(x-b%5Cright)%5E3\" alt=\"\\large 3x^2-9x+7=\\left(x-a\\right)^3-\\left(x-b\\right)^3\" align=\"absmiddle\"> para todo n&uacute;mero real x, o valor de a+b &eacute;:<\/p><p>(A) 3.<\/p><p>(B) 5.<\/p><p>(C) 6.<\/p><p>(D) 9.<\/p><p>(E) 12.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-7\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>Desenvolvendo os cubos, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3x%5E2-9x+7=%5Cleft(x-a%5Cright)%5E3-%5Cleft(x-b%5Cright)%5E3\" alt=\"\\large 3x^2-9x+7=\\left(x-a\\right)^3-\\left(x-b\\right)^3\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3x%5E2-9x+7=x%5E3-3x%5E2a+3xa%5E2-a%5E3-%5Cleft(x%5E3-3x%5E2b+3xb%5E2-b%5E3%5Cright)\" alt=\"\\large 3x^2-9x+7=x^3-3x^2a+3xa^2-a^3-\\left(x^3-3x^2b+3xb^2-b^3\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3x%5E2-9x+7=x%5E3-3x%5E2a+3xa%5E2-a%5E3-x%5E3+3x%5E2b-3xb%5E2+b%5E3\" alt=\"\\large 3x^2-9x+7=x^3-3x^2a+3xa^2-a^3-x^3+3x^2b-3xb^2+b^3\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3x%5E2-9x+7=%5Cleft(3b-3a%5Cright)x%5E2+%5Cleft(3a%5E2-3b%5E2%5Cright)x-a%5E3+b%5E3\" alt=\"\\large 3x^2-9x+7=\\left(3b-3a\\right)x^2+\\left(3a^2-3b^2\\right)x-a^3+b^3\" align=\"absmiddle\"><\/p><p style=\"text-align: left;\">O que nos leva ao seguinte sistema de equa&ccedil;&otilde;es<\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3b-3a=3\" alt=\"\\large 3b-3a=3\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3a%5E2-3b%5E2=-9\" alt=\"\\large 3a^2-3b^2=-9\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;-a%5E3+b%5E3=7\" alt=\"\\large -a^3+b^3=7\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\">\n<\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3%5Cleft(b-a%5Cright)=3\" alt=\"\\large 3\\left(b-a\\right)=3\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3%5Cleft(a%5E2-b%5E2%5Cright)=-9\" alt=\"\\large 3\\left(a^2-b^2\\right)=-9\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;-a%5E3+b%5E3=7\" alt=\"\\large -a^3+b^3=7\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\">\n<\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;b-a=1\" alt=\"\\large b-a=1\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;a%5E2-b%5E2=-3\" alt=\"\\large a^2-b^2=-3\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;b%5E3-a%5E3=7\" alt=\"\\large b^3-a^3=7\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\">\n<\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;b-a=1\" alt=\"\\large b-a=1\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft(a+b%5Cright)%5Ccdot%5Cleft(a-b%5Cright)=-3\" alt=\"\\large \\left(a+b\\right)\\cdot\\left(a-b\\right)=-3\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;b%5E3-a%5E3=7\" alt=\"\\large b^3-a^3=7\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\">\n<\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;b-a=1\" alt=\"\\large b-a=1\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft(a+b%5Cright)%5Ccdot%5Cleft(-1%5Cright)=-3\" alt=\"\\large \\left(a+b\\right)\\cdot\\left(-1\\right)=-3\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;b%5E3-a%5E3=7\" alt=\"\\large b^3-a^3=7\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\">\n<\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;b-a=1\" alt=\"\\large b-a=1\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;a+b=3\" alt=\"\\large a+b=3\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;b%5E3-a%5E3=7\" alt=\"\\large b^3-a^3=7\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\">\n<\/p><p style=\"text-align: center;\">Como a quest&atilde;o solicitou o valor de a+b, n&atilde;o &eacute; preciso ir adiante, j&aacute; temos nossa resposta: a+b=3.<\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: A<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-21\"><strong>Quest&atilde;o 21<\/strong><\/h3><p>Uma ag&ecirc;ncia de turismo vendeu um total de 78\npassagens para os destinos: Lisboa, Paris e Roma. Sabe-se que o n&uacute;mero de\npassagens vendidas para Paris foi o dobro do n&uacute;mero de passagens vendidas para\nos outros dois destinos conjuntamente. Sabe-se tamb&eacute;m que, para Roma, foram vendidas\nduas passagens a mais que a metade das vendidas para Lisboa. Qual foi o total\nde passagens vendidas, conjuntamente, para Paris e Roma?<\/p><p>(A) 26<\/p><p>(B) 38<\/p><p>(C) 42<\/p><p>(D) 62<\/p><p>(E) 68<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-8\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>Da leitura do enunciado, tiramos o seguinte sistema de equa&ccedil;&otilde;es:<\/p><p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;L+P+R=78\" alt=\"\\large L+P+R=78\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=2%5Ccdot%5Cleft(L+R%5Cright)\" alt=\"\\large P=2\\cdot\\left(L+R\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;R=2+%5Cfrac%7BL%7D%7B2%7D\" alt=\"\\large R=2+\\frac{L}{2}\" align=\"absmiddle\"><\/p><p>Dando andamento &agrave; resolu&ccedil;&atilde;o do sistema, temos:<\/p><p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;L+P+R=78\" alt=\"\\large L+P+R=78\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2L-P+2R=0\" alt=\"\\large 2L-P+2R=0\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;-L%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;+2R=4\" alt=\"\\large -L\\ \\ \\ \\ \\ \\ \\ \\ +2R=4\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\">\n<\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;L+P+R=78\" alt=\"\\large L+P+R=78\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;-3P%5C&amp;space;%5C&amp;space;%5C&amp;space;+2=-78%5Ccdot2\" alt=\"\\large -3P\\ \\ \\ +2=-78\\cdot2\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P+3R=82\" alt=\"\\large P+3R=82\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\">\n<\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;L+P+R=78\" alt=\"\\large L+P+R=78\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P+2=52\" alt=\"\\large P+2=52\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P+3R=82\" alt=\"\\large P+3R=82\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\">\n<\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;L+P+R=78\" alt=\"\\large L+P+R=78\" align=\"absmiddle\"><\/p><p style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P%5C&amp;space;%5C&amp;space;%5C&amp;space;%5C&amp;space;=52\" alt=\"\\large P\\ \\ \\ \\ =52\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;R=10\" alt=\"\\large R=10\" align=\"absmiddle\"><\/p><p>Novamente, a quest&atilde;o nos solicitou o valor da soma P+R, portanto, n&atilde;o h&aacute; necessidade de seguirmos com a resolu&ccedil;&atilde;o do sistema, uma vez que<\/p><p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P+R=52+10=62\" alt=\"\\large P+R=52+10=62\" align=\"absmiddle\"><\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: D<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-22\"><strong>Quest&atilde;o 22<\/strong><\/h3><p>Um ponto (x, y) do plano cartesiano pertence ao conjunto F se &eacute; equidistante dos eixos OX e OY e pertence ao c&iacute;rculo de equa&ccedil;&atilde;o <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5E2+y%5E2-2x-6y+2=0\" alt=\"\\large x^2+y^2-2x-6y+2=0\" align=\"absmiddle\">.&nbsp;<\/p><p>&Eacute; correto afirmar que F<\/p><p>(A) &eacute; um conjunto vazio.<\/p><p>(B) tem exatamente 2 pontos, um no primeiro quadrante e outro no segundo quadrante.<\/p><p>(C) tem exatamente 2 pontos, ambos no primeiro quadrante.<\/p><p>(D) tem exatamente 3 pontos, sendo dois no primeiro quadrante e outro no segundo quadrante.<\/p><p>(E) tem exatamente 4 pontos, sendo dois no primeiro quadrante e dois no segundo quadrante.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-9\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>O conjunto de pontos que &eacute; equidistante dos eixos coordenados &eacute; dado pelas retas<\/p><p>y = x<\/p><p>y = -x<\/p><p>Como, para pertencerem ao conjunto F, os pontos tamb&eacute;m precisam satisfazer a equa&ccedil;&atilde;o <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5E2+y%5E2-2x-6y+2=0\" alt=\"\\large x^2+y^2-2x-6y+2=0\" align=\"absmiddle\">, podemos utilizar a substitui&ccedil;&atilde;o para encontrar as coordenadas dos pontos.<\/p><p>Para o caso y = x<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5E2+x%5E2-2x-6x+2=0\" alt=\"\\large x^2+x^2-2x-6x+2=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2x%5E2-8x+2=0\" alt=\"\\large 2x^2-8x+2=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5CDelta&amp;space;=b%5E%7B2%7D-4%5Ccdot&amp;space;a%5Ccdot&amp;space;c=-82-4%5Ccdot&amp;space;2%5Ccdot&amp;space;2=64-16=48\" alt=\"\\large \\Delta =b^{2}-4\\cdot a\\cdot c=-82-4\\cdot 2\\cdot 2=64-16=48\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=%5Cfrac%7B-b%5Cpm&amp;space;%5Csqrt%7B%5CDelta&amp;space;%7D%7D%7B2a%7D=%5Cfrac%7B-(-8)%5Cpm&amp;space;%5Csqrt%7B48%7D%7D%7B2%5Ccdot&amp;space;2%7D=\" alt=\"\\large x=\\frac{-b\\pm \\sqrt{\\Delta }}{2a}=\\frac{-(-8)\\pm \\sqrt{48}}{2\\cdot 2}=\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft%5C%7B%5Cbegin%7Bmatrix%7D&amp;space;x'=%5Cfrac%7B8+4%5Csqrt%7B3%7D%7D%7B4%7D=2+%5Csqrt%7B3%7D&amp;space;&amp;&amp;space;%5C%5C&amp;space;&amp;%5C%5C&amp;space;x''=%5Cfrac%7B8-4%5Csqrt%7B3%7D%7D%7B4%7D=2-%5Csqrt%7B3%7D%5Cend%7Bmatrix%7D%5Cright.\" alt=\"\\large \\left\\{\\begin{matrix} x'=\\frac{8+4\\sqrt{3}}{4}=2+\\sqrt{3} &amp; \\\\ &amp;\\\\ x''=\\frac{8-4\\sqrt{3}}{4}=2-\\sqrt{3}\\end{matrix}\\right.\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;y=x%5Crightarrow%5Cleft(2+%5Csqrt3;2+%5Csqrt3%5Cright)\" alt=\"\\large y=x\\rightarrow\\left(2+\\sqrt3;2+\\sqrt3\\right)\" align=\"absmiddle\"><\/p><p>e<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cleft(2-%5Csqrt3;2-%5Csqrt3%5Cright)\" alt=\"\\large \\left(2-\\sqrt3;2-\\sqrt3\\right)\" align=\"absmiddle\"><\/p><p>Portanto, dois pontos do primeiro quadrante. Vejamos o pr&oacute;ximo caso.<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;y=-x\" alt=\"\\large y=-x\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x%5E2+%5Cleft(-x%5Cright)%5E2-2x-6%5Cleft(-x%5Cright)+2=0\" alt=\"\\large x^2+\\left(-x\\right)^2-2x-6\\left(-x\\right)+2=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;2x%5E2+4x+2=0\" alt=\"\\large 2x^2+4x+2=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5CDelta&amp;space;=b2-4%5Ccdot&amp;space;a%5Ccdot&amp;space;c=42-4%5Ccdot&amp;space;2%5Ccdot&amp;space;2=0\" alt=\"\\large \\Delta =b2-4\\cdot a\\cdot c=42-4\\cdot 2\\cdot 2=0\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=%5Cfrac%7B-b%5Cpm&amp;space;%5Csqrt%7B%5CDelta&amp;space;%7D%7D%7B2a%7D=%5Cfrac%7B-4%5Cpm&amp;space;0%7D%7B2%5Ccdot&amp;space;2%7D=-1\" alt=\"\\large x=\\frac{-b\\pm \\sqrt{\\Delta }}{2a}=\\frac{-4\\pm 0}{2\\cdot 2}=-1\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;y=-x%5Crightarrow&amp;space;(-1;&amp;space;1)\" alt=\"\\large y=-x\\rightarrow (-1; 1)\" align=\"absmiddle\"><\/p><p>Ou seja, um ponto no segundo quadrante.<\/p><p>At&eacute; aqui, tudo bem, temos 3&nbsp;pontos, dois no primeiro quadrante e um no segundo quadrante, o que poderia indicar a alternativa d) como nosso gabarito.<\/p><p>No entanto, a palavra &ldquo;c&iacute;rculo&rdquo;, no enunciado, remete &agrave; &aacute;rea cercada pela circunfer&ecirc;ncia. Como uma das retas (y=x) &eacute; secante &agrave; circunfer&ecirc;ncia, ter&iacute;amos infinitos pontos pertencentes ao conjunto F, n&atilde;o s&oacute; os 3 que achamos.<\/p><p>Al&eacute;m disso, a equa&ccedil;&atilde;o fornecida, n&atilde;o representa um c&iacute;rculo e sim uma circunfer&ecirc;ncia. Equa&ccedil;&otilde;es de c&iacute;rculo envolvem desigualdades, n&atilde;o igualdades.<\/p><p>Dessa forma, apesar de haver uma indica&ccedil;&atilde;o de resposta na alternativa d), indico a quest&atilde;o para anula&ccedil;&atilde;o.<\/p><p class=\"has-luminous-vivid-amber-background-color has-background\"><strong>Gabarito: PARA ANULA&Ccedil;&Atilde;O<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-23\"><strong>Quest&atilde;o 23<\/strong><\/h3><p>Uma cidade &eacute; dividida em dois Setores: o Setor Sul, com &aacute;rea de 10 km2, e o Setor Norte, com &aacute;rea de 30 km2. Ap&oacute;s um final de semana, foram divulgados os seguintes totais pluviom&eacute;tricos:<\/p><div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" width=\"331\" height=\"68\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/11\/image218.png\" alt=\"\" class=\"wp-image-34279\" srcset=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2019\/11\/image218.png 331w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2019\/11\/image218-300x62.png 300w\" sizes=\"(max-width: 331px) 100vw, 331px\" \/><\/figure><\/div><p>&Eacute; correto afirmar que o total pluviom&eacute;trico desse\nfinal de semana na cidade inteira foi de<\/p><p>(A)\n15 mm.<\/p><p>(B)\n17 mm.<\/p><p>(C)\n22 mm.<\/p><p>(D)\n25 mm.<\/p><p>(E) 28 mm.<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-10\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>O total pluviom&eacute;trico P &eacute;, na verdade, uma raz&atilde;o entre o volume de chuva que cai em uma determinada &aacute;rea, ou seja, uma raz&atilde;o entre um volume e uma &aacute;rea.<\/p><p>Seguindo essa defini&ccedil;&atilde;o, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cfrac%7B10%5C&amp;space;Km%5E2%5Ccdot%5Cleft(7+9%5Cright)%5C&amp;space;mm+30%5C&amp;space;Km%5E2%5Ccdot%5Cleft(11+17%5Cright)%5C&amp;space;mm%7D%7B10%5C&amp;space;Km%5E2+30%5C&amp;space;Km%5E2%7D\" alt=\"\\large P=\\frac{10\\ Km^2\\cdot\\left(7+9\\right)\\ mm+30\\ Km^2\\cdot\\left(11+17\\right)\\ mm}{10\\ Km^2+30\\ Km^2}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cfrac%7B%5Cleft(10%5Ccdot&amp;space;16+30%5Ccdot&amp;space;28%5Cright)%5C&amp;space;%5Ccancel%7BKm%5E2%7D%5Ccdot&amp;space;mm%7D%7B40%5C&amp;space;%5Ccancel%7BKm%5E2%7D%7D\" alt=\"\\large P=\\frac{\\left(10\\cdot 16+30\\cdot 28\\right)\\ \\cancel{Km^2}\\cdot mm}{40\\ \\cancel{Km^2}}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cfrac%7B160+840%7D%7B40%7Dmm\" alt=\"\\large P=\\frac{160+840}{40}mm\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=%5Cfrac%7B1000%7D%7B40%7Dmm\" alt=\"\\large P=\\frac{1000}{40}mm\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;P=25%5C&amp;space;mm\" alt=\"\\large P=25\\ mm\" align=\"absmiddle\"><\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: D<\/strong><\/p><h3 class=\"wp-block-heading\" id=\"h-questao-24\"><strong>Quest&atilde;o 24<\/strong><\/h3><figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/11\/image219.png\" alt=\"\" class=\"wp-image-34282\"><\/figure><p>As poss&iacute;veis solu&ccedil;&otilde;es, em polegadas (inches, em ingl&ecirc;s), para o problema matem&aacute;tico proposto no quadrinho, no caso em que os pontos A, B e C est&atilde;o em uma mesma reta, s&atilde;o<\/p><p>(A) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B10%7D%7B3%7D\" alt=\"\\large \\frac{10}{3}\" align=\"absmiddle\"> e 10<\/p><p>(B) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B10%7D%7B3%7D\" alt=\"\\large \\frac{10}{3}\" align=\"absmiddle\">, 5 e 10<\/p><p>(C) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B5%7D%7B3%7D,%5C&amp;space;%5C&amp;space;%5Cfrac%7B10%7D%7B3%7D\" alt=\"\\large \\frac{5}{3},\\ \\ \\frac{10}{3}\" align=\"absmiddle\"> e 10<\/p><p>(D) <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B5%7D%7B3%7D\" alt=\"\\large \\frac{5}{3}\" align=\"absmiddle\"> e 10<\/p><p>(E)&nbsp;<img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;%5Cfrac%7B10%7D%7B3%7D\" alt=\"\\large \\frac{10}{3}\" align=\"absmiddle\"> e 5<\/p><h3 class=\"wp-block-heading\" id=\"h-resolucao-comentada-11\"><strong>Resolu&ccedil;&atilde;o Comentada<\/strong><\/h3><p>O texto deixa claro que a dist&acirc;ncia entre <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;A\" alt=\"\\large A\" align=\"absmiddle\"> e <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;C(d_%7BAC%7D)\" alt=\"\\large C(d_{AC})\" align=\"absmiddle\"> &eacute; o dobro da dist&acirc;ncia entre <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;A\" alt=\"\\large A\" align=\"absmiddle\"> e <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;B(d_%7BAC%7D)\" alt=\"\\large B(d_{AC})\" align=\"absmiddle\"> ou seja<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_%7BAC%7D=2%5Ccdot&amp;space;d_%7BAB%7D\" alt=\"\\large d_{AC}=2\\cdot d_{AB}\" align=\"absmiddle\"><\/p><p>Utilizando um sistema ordenado e respeitando a ordem de que A, B e C s&atilde;o colineares, podemos representar os pontos em uma reta orientada.<\/p><div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/11\/image220.png\" alt=\"\" class=\"wp-image-34286\"><\/figure><\/div><p>Dessa forma, seguindo as orienta&ccedil;&otilde;es do enunciado, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_%7BAC%7D=2%5Ccdot&amp;space;d_%7BAB%7D\" alt=\"\\large d_{AC}=2\\cdot d_{AB}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x-0=2%5Ccdot%5Cleft(5-x%5Cright)\" alt=\"\\large x-0=2\\cdot\\left(5-x\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=10-2x\" alt=\"\\large x=10-2x\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;3x=10\" alt=\"\\large 3x=10\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=%5Cfrac%7B10%7D%7B3%7D\" alt=\"\\large x=\\frac{10}{3}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_%7BAC%7D=%5Cfrac%7B10%7D%7B3%7D\" alt=\"\\large d_{AC}=\\frac{10}{3}\" align=\"absmiddle\"><\/p><p>At&eacute; aqui, tudo bem. No entanto, h&aacute; uma segunda possibilidade para posicionarmos o ponto A, fora do intervalo BC, veja:<\/p><div class=\"wp-block-image\"><figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/cdn.blog.estrategiavestibulares.com.br\/vestibulares\/wp-content\/uploads\/2019\/11\/image221.png\" alt=\"\" class=\"wp-image-34287\"><\/figure><\/div><p>Com essa nova representa&ccedil;&atilde;o, mas ainda com a premissa do exerc&iacute;cio de que <img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_%7BAC%7D=2%5Ccdot&amp;space;d_%7BAB%7D\" alt=\"\\large d_{AC}=2\\cdot d_{AB}\" align=\"absmiddle\">, temos:<\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_%7BAC%7D=2%5Ccdot&amp;space;d_%7BAB%7D\" alt=\"\\large d_{AC}=2\\cdot d_{AB}\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x-0=2%5Ccdot%5Cleft(x-5%5Cright)\" alt=\"\\large x-0=2\\cdot\\left(x-5\\right)\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;x=2x-10\" alt=\"\\large x=2x-10\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;10=x\" alt=\"\\large 10=x\" align=\"absmiddle\"><\/p><p><img decoding=\"async\" src=\"https:\/\/latex.codecogs.com\/svg.latex?%5Clarge&amp;space;d_%7BAC%7D=10\" alt=\"\\large d_{AC}=10\" align=\"absmiddle\"><\/p><p>Assim, temos duas possibilidades para a posi&ccedil;&atilde;o do ponto A, nas condi&ccedil;&otilde;es dadas.<\/p><p class=\"has-light-green-cyan-background-color has-background\"><strong>Gabarito: A<\/strong><\/p><p>&Eacute; isso, pessoal! Espero que tenham curtido a resolu&ccedil;&atilde;o da prova de Matem&aacute;tica da prova do Vestibular FUVEST 2020.&nbsp;Sigam-me nas redes sociais. T&ecirc;m muitas dicas l&aacute;. Mande uma mensagem, caso tenha tido alguma d&uacute;vida. Abra&ccedil;os!<\/p><p><strong>Instagram:<\/strong>&nbsp;<a href=\"https:\/\/www.instagram.com\/professor.marcal\/\" target=\"_blank\">@professor.marcal<\/a><\/p><p>\n\n\n\n\n\n<a id=\"cta\" class=\"cta-imagem\" href=\"https:\/\/vestibulares.estrategia.com\/curso\/pacote-para-fuvest\/\" target=\"blank\">\n                <img decoding=\"async\" width=\"100%\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2021\/12\/Fuvest.jpg\" alt=\"CTA Curso Fuvest\" title=\"CTA Curso Fuvest\">\n        <\/a>\n\n\n\n<\/p><h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Veja-tambem\"><\/span>Veja tamb&eacute;m:<span class=\"ez-toc-section-end\"><\/span><\/h2><ul class=\"wp-block-list\"><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/geometria\/\">Geometria: o que &eacute; e como &eacute; classificada<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/figuras-geometricas\/\">Figuras Geom&eacute;tricas: planas, espaciais e como aparece nos vestibulares<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/geometria-plana-no-enem\/\">Geometria Plana no Enem: conceito e defini&ccedil;&otilde;es<\/a><\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/semelhanca-de-triangulos\/\">Semelhan&ccedil;a de tri&acirc;ngulos<\/a><\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/geometria-analitica\/\">Geometria anal&iacute;tica: f&oacute;rmulas, gr&aacute;ficos e equa&ccedil;&otilde;es<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/geometria-espacial\/\">Geometria Espacial: conhe&ccedil;a os s&oacute;lidos e f&oacute;rmulas<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/progressao-aritmetica\/\">Progress&atilde;o aritm&eacute;tica: o que &eacute;, f&oacute;rmulas, usos e exemplos<\/a><\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/prova-matematica-2-fase-unicamp-2020\/\">Resolu&ccedil;&atilde;o da prova de Matem&aacute;tica da 2&ordf; Fase da Unicamp 2020<\/a>&nbsp;<\/li><li><a href=\"https:\/\/vestibulares.estrategia.com\/portal\/matematica\/prova-unesp-2020-matematica\/\">Prova UNESP 2020 &ndash; Matem&aacute;tica &ndash; Resolu&ccedil;&atilde;o Comentada<\/a><\/li><\/ul><\/p>\n","protected":false},"excerpt":{"rendered":"Ol&aacute;, pessoal&hellip; Tudo bem? Sou o prof. Mar&ccedil;al, do Estrat&eacute;gia Vestibulares, e escrevo este artigo para resolver as&hellip;\n","protected":false},"author":18,"featured_media":34289,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[29],"tags":[],"wl_entity_type":[732],"class_list":{"0":"post-34248","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-matematica","8":"wl_entity_type-article"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v25.9 (Yoast SEO v25.9) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Prova de matem\u00e1tica Fuvest 2020 com Resolu\u00e7\u00e3o Comentada<\/title>\n<meta name=\"description\" content=\"Voc\u00ea teve dificuldade nas quest\u00f5es da prova de Matem\u00e1tica Fuvest 2020? 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